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Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

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Abstract

We use the method of Lie symmetry analysis to investigate the properties of a (2+1)-dimensional KdV–mKdV equation. Using the Ibragimov method, which relies only on the existence of the commutator table, we construct an optimal system of one-dimensional subalgebras of the Lie algebra and study invariant solutions and similarity reductions by considering representatives of the optimal system. To analyze some nonlocal symmetry properties, we apply the truncated Painlevé expansion method and obtain two Bäcklund transformations that are not autotransformations and one auto-Bäcklund transformation. To localize the nonlocal symmetry and obtain a local Lie point symmetry, we introduce an expanded system. Using solutions of the corresponding Cauchy problems for Lie point symmetries, we prove a theorem on a finite symmetry transformation and find the \(n\)th Bäcklund transformation in terms of determinants. Based on one of the obtained Bäcklund transformations that are not autotransformations, we derive lump-type solutions. In addition, we prove the integrability of the equation by the consistent Riccati expansion method. We present explicit soliton-cnoidal wave solutions and investigate the dynamical characteristics of the obtained solutions using numerical analysis.

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Funding

This research is supported by Shanxi Province Science Foundation for Youths (No. 201901D211274), Research Project Supported by Shanxi Scholarship Council of China (No. 2020-105), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2019L0531), and Fund for Shanxi “1331KIRT.”

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Authors and Affiliations

  1. Department of Mathematics, North University of China, Taiyuan, Shanxi, China

    Zhonglong Zhao

  2. College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan, Shanxi, China

    Lingchao He

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  1. Zhonglong Zhao
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  2. Lingchao He
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Correspondence to Zhonglong Zhao.

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Zhao, Z., He, L. Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation. Theor Math Phys 206, 142–162 (2021). https://doi.org/10.1134/S0040577921020033

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